Problem: What is the value of $\dfrac{d}{dx}\csc(x)$ at $x=\dfrac{3\pi}{2}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $-1$ (Choice B) B $1$ (Choice C) C $0$ (Choice D) D $\dfrac{1}{2}$
Answer: Let's first find $\dfrac{d}{dx}\csc(x)$. Then, we can evaluate it at $x=\dfrac{3\pi}{2}$. Recall that the derivative of $\csc(x)$ is $-\dfrac{\cos(x)}{\sin^2(x)}$, or $-\csc(x)\cot(x)$. Put another way, $\dfrac{d}{dx}[\csc(x)]=-\dfrac{\cos(x)}{\sin^2(x)}=-\csc(x)\cot(x)$. [Is there a way to know this without memorizing?] Now let's plug in $x={\dfrac{3\pi}{2}}$ : $\begin{aligned} &\phantom{=}-\dfrac{\cos\left({\dfrac{3\pi}{2}}\right)}{\sin^2\left({\dfrac{3\pi}{2}}\right)} \\\\ &=-\dfrac{0}{\left(-1\right)^2} \\\\ &=0 \end{aligned}$ In conclusion, the value of $\dfrac{d}{dx}\csc(x)$ at $x=\dfrac{3\pi}{2}$ is $0$.